A hot body placed in air is cooled down according to Newton's law of cooling, the rate of decrease of temperature being `k` times the temperature difference from the surrounding. Starting from `t=0` , find the time in which the body will loss half the maximum heat it can lose.

Explanation

We have,
`(d theta)/(dt)=-k(theta-theta_(0))`
where `theta_(0)` is the temperature of the surrounding and `theta` is the temperature of the body at time `t`. Suppose `theta=theta_(1)` at `t=0`
Then,
`int_(theta_(1))^(theta)(d theta)/(theta-theta_(0))= -kint_(0)^(t)dt`
or `In(theta-theta_(0))/(theta_(1)-theta_(0))=-kt`
or, `theta-theta_(0)=(theta_(1)-theta_(0))e^(-kt)`.........`(i)`
The body continues to lose heat till its temperature becomes equal to that of the surrounding. the loss of heat in this entire period is
`DeltaQ_(m)=ms(theta_(1)-theta_(0))/(2)`
If the body loses this heat in time `t_(1)`, the temperature at `t_(1)` will be
`theta_(1)-(theta_(1)-theta_(0))/(2)=(theta_(1)+theta_(0))/(2)`
Putting these value of time and temperature in `(i)`,
`(theta_(1)+theta_(0))/(2)-theta_(0)=(theta_(1)-theta_(0))e^(-kt)`
or, `e^(-kt)=(1)/(2)`
or `t_(1)=(In2)/(k)`